I believe that inside the lore of world of warcraft, no one ever stated that someone from any race can't start practicing a form of any kind of class. For instance, if a troll really wanted to know how to be a mage, he could and he can! In this case, unless there's something preventing undead from being something, they can be anything. Even druids. Besides, some undead are reincarnated paladins.

Undead Paladins would feel a "burning" when using the holy power, much the same way Holy undead priests feel it. They would not be filled with light, infact they would be more like the Blood Paladins who bend the light to their will, rather than channel it. Sun walkers (Tauren Pallies) are not Paladins either, they use the power of the sun the same way druids use the moon. But for the purpose of Gameplay, they are classified as Paladins. So, Undead Paladins would be HUMAN paladins who were killed and resurrected by the Forsaken and still have a knowledge of the light, but now bend it to their will rather than channel it. It has been done already:

While Holy magic does burn undead, the Undead can still wield it. According to Blizzard, any Forsaken who is wielding the Light causes themselves great pain. Some Undead still worship the Light, and we have seen a couple of Undead paladins who still use it to their advantage. I think it'd be interesting to see Undead paladins in the Horde, personally.

Following the fall of the Lich King the ghost of Sir Zeliek reappeared in his homeland (somewhere in Lordaeron?). It began appearing to certain undead warriors and priests much like how Uther's ghost appears in a few quests. He sought to help redeem some of the Forsaken and guide them back towards righteousness. A new order was formed following his example and they become a bridge between the Forsaken and the Argent Crusade.

Okay, that's not much but it could work. So long as someone comes up with a reason for why/how Sylvanas doesn't squash them before they get too powerful.

There is an explanation about how Undead can be Holy Priests - which is that they suffer whilst channeling the light, because it burns them.

The undead, undead of any kind, are forsaken by the Light. But, like how Goblins "bribe" the elements to do their bidding(most of them, anyway), the Forsaken could possibly channel/will it by force, although the pain would still be inevitable. So Undead paladins, whilst a valid option, would suffer as much as a Priest, if not more.

Yes it is possible. It causes them pain, but they can use the Holy Light, just because someone is undead does not mean that they must be evil. remember undeath was forced upon most people, they can remain the person they were in life, not all go dark side

There is only one god and his name is Sargeras.

Tell them only that the Lich King is dead and that World of Warcraft died with him.